Finite Group Databasebeta

← C3×#2076 #72791

Subgroups — C3×#2076 #72791

Order 384.

256 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#107C211yesmaximal, characteristic, complemented
2C2×C3×C22⋊C8C211yesmaximal, characteristic
3C3×#380C212yesmaximal
4C3×#340C212yesmaximal, complemented
5C3×#340C211yesmaximal, characteristic, complemented
6#2076C311yesmaximal, characteristic, direct product factor, Sylow
7C23×C3×C4C2211yescharacteristic
8C22×C3×C8C2211yescharacteristic
9C22×C3×C8C2211yescharacteristic
10C2×C3×C4⋊C4C2212yes—
11C2×C3×C4⋊C4C2212yes—
12C2×C3×C22⋊C4—22no—
13C3×#107—22no—
14C3×C22⋊C8—22no—
15C3×#95—22no—
16C2×#107C2×C311yescharacteristic, complemented
17C2×C22⋊C8C2×C311yescharacteristic
18#380C2×C312yes—
19#340C2×C311yescharacteristic, complemented
20#340C2×C312yescomplemented
21C24×C3D411yescharacteristic
22C22×C3×C4—21no—
23C22×C3×C4—21no—
24C22×C3×C4—22no—
25C22×C3×C4—22no—
26C22×C3×C4—22no—
27C22×C3×C4—22no—
28C22×C3×C4D411yescharacteristic
29C22×C3×C4C2311yescharacteristic
30C2×C3×C8—22no—
31C2×C3×C8—21no—
32C2×C3×C8—22no—
33C2×C3×C8D412yes—
34C3×C4⋊C4—22no—
35C3×C4⋊C4—22no—
36C3×C4⋊C4—22no—
37C3×C4⋊C4—22no—
38C3×C22⋊C4—44no—
39C23×C4C22×C311yescharacteristic
40C22×C8C22×C311yescharacteristic
41C22×C8C22×C311yescharacteristic
42C2×C4⋊C4C22×C312yes—
43C2×C4⋊C4C22×C312yes—
44C2×C22⋊C4—22no—
45#107—22no—
46C22⋊C8—22no—
47#95—22no—
48C23×C3—21no—
49C23×C3—22no—
50C23×C3—21no—
51C23×C3—22no—
52C23×C3Q1612yes—
53C23×C3C2×D411yescharacteristic
54C2×C3×C4—21no—
55C2×C3×C4—22no—
56C2×C3×C4—41no—
57C2×C3×C4—41no—
58C2×C3×C4—42no—
59C2×C3×C4—42no—
60C2×C3×C4—42no—
61C2×C3×C4—42no—
62C2×C3×C4—42no—
63C2×C3×C4—42no—
64C2×C3×C4—42no—
65C2×C3×C4C2×D411yescharacteristic
66C2×C3×C4#4112yes—
67C2×C3×C4#4112yes—
68C2×C3×C4#4111yescharacteristic
69C3×C8—22no—
70C3×C8—41no—
71C24C3×D411yescharacteristic
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—22no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4C3×D411yescharacteristic
79C22×C4C23×C311yescharacteristic
80C2×C8—22no—
81C2×C8—21no—
82C2×C8—22no—
83C2×C8C3×D412yes—
84C4⋊C4—22no—
85C4⋊C4—22no—
86C4⋊C4—22no—
87C4⋊C4—22no—
88C22⋊C4—44no—
89C22×C3—22no—
90C22×C3—24no—
91C22×C3—41no—
92C22×C3—41no—
93C22×C3—42no—
94C22×C3#12111yescharacteristic
95C22×C3#12312yes—
96C22×C3C2×Q1611yescharacteristic
97C22×C3#13511yescharacteristic
98C22×C3AGL(1,ℤ/8)12yes—
99C3×C4—41no—
100C3×C4—82no—
101C3×C4—82no—
102C3×C4#12311yescharacteristic
103C3×C4#12411yescharacteristic
104C3×C4#12612yes—
105C23—21no—
106C23—22no—
107C23—21no—
108C23—22no—
109C23C3×Q1612yes—
110C23C2×C3×D411yescharacteristic
111C2×C4—21no—
112C2×C4—22no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4C2×C3×D411yescharacteristic
123C2×C4C3×#4112yes—
124C2×C4C3×#4112yes—
125C2×C4C3×#4111yescharacteristic
126C8—22no—
127C8—41no—
128C2×C3—42no—
129C2×C3#39711yescharacteristic
130C2×C3#46711yescharacteristic
131C2×C3#46911yescharacteristic
132C2×C3#48212yes—
133C2×C3#48412yes—
134C22—22no—
135C22—24no—
136C22—41no—
137C22—41no—
138C22—42no—
139C22C3×#12111yescharacteristic
140C22C3×#12312yes—
141C22C2×C3×Q1611yescharacteristic
142C22C3×#13511yescharacteristic
143C22C3×AGL(1,ℤ/8)12yes—
144C4—41no—
145C4—82no—
146C4—82no—
147C4C3×#12311yescharacteristic
148C4C3×#12411yescharacteristic
149C4C3×#12612yes—
150C3#207611yescharacteristic, direct product factor, Sylow
151C2—42no—
152C2C3×#39711yescharacteristic
153C2C3×#46711yescharacteristic
154C2C3×#46911yescharacteristic
155C2C3×#48212yes—
156C2C3×#48412yes—
subgroupG/H
center, socleC23×C3C2×D4
commutator, Frattini, cosocleC22×C4C23×C3
prime radicalC24×C3D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C3×#107C2semidirect
4. C3×#340C2semidirect
5. C3×#340C2semidirect
6. #2076C3direct (N × H)
16. C2×#107C2×C3semidirect
19. #340C2×C3semidirect
20. #340C2×C3semidirect
150. C3#2076direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×#1072
C2×C3×C22⋊C82
C3×#3802
C3×#3402
#20763